• Corpus ID: 51853516

A Reidemeister-Schreier theorem for finitely $L$-presented groups

  title={A Reidemeister-Schreier theorem for finitely \$L\$-presented groups},
  author={Ren{\'e} Hartung},
  journal={arXiv: Group Theory},
  • René Hartung
  • Published 1 August 2011
  • Mathematics
  • arXiv: Group Theory
We prove a variant of the well-known Reidemeister-Schreier theorem for finitely $L$-presented groups. More precisely, we prove that each finite index subgroup of a finitely $L$-presented group is itself finitely $L$-presented. Our proof is constructive and it yields a finite $L$-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely $L$-presented group to be invariantly $L$-presented itself. 

Tables from this paper

A Note on Invariantly Finitely $L$-Presented Groups
In the first part of this note, we introduce Tietze transformations for $L$-presentations. These transformations enable us to generalize Tietze's theorem for finitely presented groups to invariantly
  • M. Benli
  • Mathematics
    Glasgow Mathematical Journal
  • 2011
Abstract In this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely
Investigating self-similar groups using their finite $L$-presentation
Self-similar groups provide a rich source of groups with interesting properties; e.g., infinite torsion groups (Burnside groups) and groups with an intermediate word growth. Various self-similar
Presentations and Structural Properties of Self-similar Groups and Groups without Free Sub-semigroups
This dissertation is devoted to the study of self-similar groups and related topics. It consists of three parts. The first part is devoted to the study of examples of finitely generated amenable


Coset Enumeration for certain Infinitely Presented Groups
This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable.
A finitely generated, infinitely related group with trivial multiplicator
  • G. Baumslag
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1971
We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator. 1. The purpose of this note is to establish the exitense of a finitely generated group which
A Nilpotent Quotient Algorithm for Certain Infinitely Presented Groups and its Applications
A nilpotent quotient algorithm is described for a certain class of infinite presentations: the so-called finite L-presentations and conjectural descriptions of the lower central series structure of various interesting groups including the Grigorchuk supergroup, the Brunner–Sidki–Vieira group, the Basilica group, and certain generalizations of the Fabrykowski–Gupta group are obtained.
A Reidemeister-Schreier Program
The Reidemeister-Schreier method yields a presentation for a subgroup H of a group G when H is of finite index in G and G is finitely presented. This paper describes the implementation and
Applications of computational tools for finitely presented groups
Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in stand-alone programs and in more
Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients
We describe an algorithm for computing successive quotients of the Schur multiplier M ( G ) of a group G given by an invariant finite L -presentation. As applications, we investigate the Schur
A nilpotent quotient algorithm for L-presented groups
The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The
A practical method for enumerating cosets of a finite abstract group
An important problem in finite-group theory is the determination of an abstract definition for a given group , that is, a set of relations between k generating operations S 1 , …., S k of , such that
On the failure of the co-hopf property for subgroups of word-hyperbolic groups
We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic groupG such thatH is one-ended, andH does not split over a cyclic group, andH is isomorphic to one of its
On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are